We're learning about this in chemistry right now. What happens in 3D goes like this:

2 pairs align themselves along a line
3 pairs point at the corners of an equilateral triangle
4 pairs point at the corners of a regular tetrahedron
5 pairs have the equilateral triangle plus the line, perpendicular to each other
6 pairs have three lines, each perpendicular to each other

Now what I'd like to know is if you can do more than 6 pairs in 3D, and what happens in 4D, after 5 pairs (2-4 are the same, 5 is obviously going to point at the corners of a regular pentachoron).

There are four pairs of electrons around the C, and they want to be as far away from each other as possible, because electrons repel. But the protons in hydrogen want to still be close to their electrons. So what happens is you get the carbon atom in the middle, with the four hydrogen atoms at the corners of an imaginary tetrahedron (centered at the carbon atom).

iNVERTED wrote:We're learning about this in chemistry right now. What happens in 3D goes like this:

2 pairs align themselves along a line3 pairs point at the corners of an equilateral triangle4 pairs point at the corners of a regular tetrahedron5 pairs have the equilateral triangle plus the line, perpendicular to each other6 pairs have three lines, each perpendicular to each other

For four dimensions one can set any number of points up in this way, by starting off with a p×p bi-prism, and then (for some a), mark off 1,a 2,2a, 3,3a &c. This will give a uniformly distributed set of points. One also has